Abelian Groups Review

Abelian Groups Review

We will now review some of the recent material regarding abelian groups.

• Recall from the Abelian Groups page that if $(G, *)$ is a group and for all $a, b \in G$ we have that $a * b = b * a$, i.e., the operation $*$ is commutative, then $(G, *)$ is called an Abelian Group or a Commutative Group.
• We noted that many of the groups we have already seen are indeed abelian groups. For example, $(\mathbb{R}, +)$ and $(\mathbb{Z}, +)$ are indeed abelian groups. However, not all groups are abelian, and the prime example that we looked at was the set of $2 \times 2$ matrices with the operation $*$ of matrix multiplication. We found an example of two matrices $A$ and $B$ such that $A * B \neq B * A$.
• On the Basic Theorems Regarding Abelian Groups page we looked at a couple of simple theorems regarding abelian groups. We first looked at a criterion for determining if a group $(G, *)$ was abelian. We saw that if for all $a, b \in G$ we have that $(a * b)^2 = a^2 * b^2$ then $(G, *)$ is an abelian group.
• We also saw that if for all $a \in G$ we have that $a = a^{-1}$ then $(G, *)$ is an abelian group.
• On the The Center of a Group page we defined an important notion for groups. We said that the Center of a group $(G, *)$ denoted $Z(G)$ is defined to be the set of all elements in $G$ that commute with every element in $G$. That is:
(1)
\begin{align} \quad Z(G) = \{ a \in G : \mathrm{for} \: \mathrm{all} \: g \in G, \: a * g = g * a \} \end{align}
• We then proved that a group $(G, *)$ is abelian if and only if $Z(G) = G$ which intuitively makes sense. We also proved that $(Z(G), *)$ is itself an abelian subgroup of $(G, *)$.
• We then went through a bunch of examples of abelian groups which are linked below: